15. THE GOLDEN AGE AND THE GOLDEN MEAN

The second half of the 5th century B.C. was the Golden Age of Greece. This was the
period of her most beautiful art and architecture, and some of her wisest thinkers
besides. Both owed much to the popular new study of geometry.

By the start of the next century, geometry itself was entering its own classic age with
a series of great developments, including the Golden Mean. The times were glorious in
many ways. The Persian invaders had been driven out of Hellas forever, and Pericles
was rebuilding Athens into the most beautiful city in the world. At his invitation, Greek
mathematicians from elsewhere flocked into the new capital. From Ionia came Anaxa-
goras, nicknamed "the mind." From southern Italy and Sicily came learned Pythagoreans
and the noted Zeno of Elea. And their influence was felt over all Athens.

High on the hill of the Acropolis rose new marble temples and bronze and painted
statues. Crowds thronged the vast new open-air theater nearby, to hear immortal
tragedies and comedies by the greatest Greek playwrights. These splendid public works
were completed under the direction of the sculptor Phidias and several architects, all of
whom knew and used the principles of geometry and optics. "Success in art," they
insisted, "is achieved by meticulous accuracy in a multitude of mathematical
proportions." And their buildings had a dazzling perfection never seen before-the beauty
of calculated geometric harmony.

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Elsewhere in the city, the impact of the new geometry took another form. On the
narrow streets of Athens walked world-famous philosophers, talking to the people,
lecturing on mathematics, geography, rhetoric, how to live the good life. Socrates and
others asked, "What is beauty? What is virtue?"- and tried to teach men to think out the
answers.

Their method was borrowed from the geometers. They called it dialectics, and it was
patterned after the deductive reasoning and proofs of geometry. "For geometry," they
said, "will lead the soul toward truth and create the spirit of philosophy."

And geometry itself made tremendous strides in the Golden Age and the darker time
that followed. Even after Athenian democracy collapsed in the war with Sparta, geometry
continued to flourish in the Athens of the restored aristocracy.

But now, in the 4th century, the study was carried on in schools with grounds and
buildings of their own. The first and most famous of these was the Academy, headed by
the great philosopher Plato. It was located in an olive grove a half-mile outside of town,
and over ifs gate was this inscription:

LET NONE IGNORANT OF GEOMETRY ENTER HERE

Plato's Academy was the earliest institution of higher learning. Its curriculum was
frankly inspired by the old program of the Secret Brotherhood. Studies were broader now-
the highest branch was moral and political philosophy. But the ideal was still pure
wisdom, and the basic training was still in the "Mathemata." Plato was partly a
Pythagorean.

When his teacher, Socrates, was put to death by the Athenian government, Plato had
fled to Sicily. There he studied mathematics under noted Pythagoreans, picked up
mystical ideas, and

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dabbled in aristocratic politics. Finally, he came home to Athens to found his own school
and make it the great mathematical center of the Greek world. Most of the
mathematicians of that era were his friends or associated with his Academy.

Perhaps the most gifted geometer to study there was Eudoxus of Cnidus, who finally
broke the deadlock of the irrationals, and freed geometry for the great advances that
were to come. How he did this-with his work on the Golden Mean and his new theory of
proportion-is an exciting story. And if we add a bit of imagination, it gives us a
fascinating glimpse of Athens and the Academy in Plato's time.

At the age of twenty-four, Exodus came to Athens from his home town of Cnidus on
the Black Sea, in order to study at Plato's Academy. He was so poor that he could not
afford lodgings in the city, but lived in the small seaport of Piraeus and walked to school
every day. Of course, he had already studied some geometry; it was the entrance
requirement. But at the Academy he got particularly interested in the matter of an
irrational number of a geometric figure. For in Athens the problem was in plain sight
every day, in a concrete, or rather, a marble form.

On the high Acropolis, against the shimmering sky, stood the beautiful temple called the
Parthenon-the most wonderful monument of the Age of Pericles, the "perfect" building
whose ruins enthrall us even today. The Parthenon had been designed by Ictinus and
Callicrates according to mathematical principles Its surrounding pillars were an example
of "number" applied: 8 pillars in front, an even number, as Pythagoras had advised, so
no central posts would block the view; but 17 pillars on each side, where it was all right
to have an odd number. And some of

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its lines were deliberately curved and slanted to correct optical distortions.

But above all, the Parthenon was a crowning example of proportion in architecture.
Scholars still marvel at the logical and harmonious ratios in the whole building
and its various parts. And this beauty was achieved with one of the "dynamic
rectangles" then in vogue.

Like many Greek temples of time, the Parthenon used the "root five rectangle," a
rectangle with an irrational side the square root of 5. How did this root five
rectangle come to be used? How was it constructed and shown to be irrational? How did
Eudoxus analyze in it the most beautiful of all linear proportions, the Golden Section, or
Golden Mean? That is our story.

The development was natural in the architecture of the Golden Age. Greek builders, we
must remember, did not have a minutely

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graduated measuring rod, in inches or centimeters, like ours. Ground plans were
still laid out in the old way, with string (rope), straightedge, level, and carpenter's right
angle or "set square." And some of the older temples, and even a few new ones, were
quite carelessly designed.

But as geometry became popular in Athens, architects took to drawing careful plans
with string and straightedge, for geometric constructions could be enlarged easily and
accurately in the building itself.

Temples remained severely rectangular, but now the favorite rectangle was made by
a "construction": a square inscribed in a semicircle. This figure gave you the shape of
the rectangle: it was as long as the semicircle's diameter, as high as the
inscribed square. Calculating its numerical dimensions was easy with the Pythagorean
theorem; any builder or Academy student knew how in those days. The rectangle had an
irrational dimension. When its width was 1, its length was the square root of 5.

This "root five rectangle" was enough to discourage any member of the Secret
Brotherhood-but Eudoxus belonged to a new age. After studying for a while at the
Academy, he went to Egypt, where he studied under the learned priests. Afterward, he
traveled and established his own school. Then, years later, he returned to Athens to
revisit his former master Plato.

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This time, Eudoxus arrived not as a poor student, but an acknowledged master of
geometry. In token of his importance, he now wore his beard and eyebrows shaved in
the Egyptian manner. he was accompanied by several of his own disciples. A holiday in
his honor was declared at the Academy. All the students wanted to see him, and they
crowded into the famous open-air lecture space shaded by the grove of olive trees. And
there-we may imagine-he gave them the geometric solution of the proportion in the "root
five rectangle," which had puzzled him as a student.

Eudoxus chose his words with care. He had promised to tell Plato a great discovery
at dinner, and it would be based on this novel demonstration.

"I will ask you," he said, "to disregard numbers entirely, and forget all about the
numerical dimensions of the 'root five rectangle.' We will try instead to find a proportion
among the geometric quantities. So now, look at the construction itself, the square
inscribed in the semicircle." Using string and straightedge, Eudoxus drew it on the
sand.

"Look at the straight lines in the whole construction. You will see that there are only two
geometric quantities throughout. What are these? One is b, the width of the
square, wherever it occurs. Now study the diameter of the semicircle. On that line
there are three segments. The long segment is simply b, the base of the
square. The two short segments, a and a, on either side, are equal-for each
equals the radius minus 1/2 the base of the square.

"Reduced to its simplest terms, therefore, the problem is to find a proportion
between the geometric quantities a and b, irrespective of any numerical
dimensions. So here is the figure once more, simplified to show the problem in this
simplest form.

Consider only the line

that is, only that portion of the diameter, where our two quantities a and b
can be defined as a short and a long segment of one line.

"Now here is my question: What is the proportion that links a and b,
the short and long segments of this line? Does anybody see how to find out?"

A ripple of excitement rose from the students gathered in the grove of the Academy,
as they peered at the diagram and discussed this "simple" problem in whispers. Plato
himself stood by, smiling. Finally, when no one volunteered, Eudoxus raised his hand
for attention and continued.

"Nothing could be simpler than the answer. It involves a very easy construction that
you all know already. From the upper right corner of the square, I will just draw two lines
to the ends of the diameter. What does that give you?" He pointed to an eager student in
the front row.

"A right triangle, of course," the boy almost shouted. "Lines drawn from any point
on the circumference to the ends of the diameter make a right angle!"

"What else do you see?"

Several students answered at once: "Inside this large right triangle are two other right
triangles! They are formed by a side of the square-but it is now a perpendicular line
dropped from the vertex of the large right triangle to its hypotenuse."

"Absolutely right!" said Eudoxus, pointing them out. "We will call one S for Small,
and the other M for Medium; and the large right triangle, of course, can be L for
Large. Now, do you see any relationship between these three right triangles?"

There was a pause, while all the students stared intently at the diagram. Suddenly a
boy called out from the back row, "The three right triangles are similar, aren't they?"

"How do you prove that?" Eudoxus was nodding his approval.

"Sir, they are similar because their angles are equal. If you will kindly spin the three
right triangles around and draw them side by side and upright, then everyone else can
see the proof."

Eudoxus gladly obliged, and, using his pointer, he explained for the benefit of the
slower students, "Notice on the figure that each of the smaller triangles has an angle in
common with the large triangle. But we know that in any right triangle the sum

of the other two angles is 90*. So each of the remaining angles must be equal
respectively. The three right triangles are therefore similar, just as-what is your name,
lad?-just as Meno here has said, because their angles are equal. Meno has solved the
problem!"

"But sir," protested Meno in amazement, "of what use is it for us to know that the right
triangles are similar?"

"Of what use?" repeated Eudoxus, laughing. "Look again, all of you, and you will see
the beautiful proportion that links the geometric quantities a and b." He
pointed in quick succession to all the drawings on the sand.

"Just take the dimensions from the final figure, and put them on the easy-to-see
similar right triangles, just the Small and Medium ones. You know that when right
triangles are similar, their corresponding sides are in proportion. Therefore,

Short Side of Small Triangle is to Long Side of Small Triangle as

Short Side of Medium Triangle is to Long Side of Medium Triangle or in
other words: a is to b as b is to a + b

"That is your proportion! Just read it off on the line, and you will see how beautiful
it is:

"High-rete! High-rete! High-rete!" cried all the students in unison-the Greek equivalent
of three cheers.

Plato himself joined the chorus of praise and made a short speech: "You have just seen
a beautiful demonstration and proof-one of the most ingenious in geometry. This
proportion is far more significant than the problem that led to it. So I will ask you all to
review the construction for your next assignment.

"By inscribing a square in a semicircle, you can do something truly marvelous with a
straight line. You can divide that line into two unequal sections, in such a way that the
short section is to the long section as the long section is to the whole line. Do you
appreciate this proportion? You are thus dividing a line geometrically into its extreme
and mean proportional. This section, or cutting, of a line is so important that from now on
we will call it THE SECTION." Plato drew and wrote on the sand.

Plato gave a banquet in Eudoxus' honor that night-history records the event-and heard
the rest of the discovery from Eudoxus' own lips. Before we join them at dinner, let us
pause (like the boys at the Academy) to appreciate the importance of "The Section." Plato
himself, in his writings, always called it that. But later writers named it the Golden
Section or the Golden Mean.

The lasting fame of the Golden Mean rests not only on the sheer beauty of the
proportion itself, but on its use in architecture and art. The "root five" and Golden Section
rectangles were used frequently in Greek buildings. Scholars have since found that
many of the loveliest classical vases and statues cherished today, on the hills of Greece
and in museums throughout the world, are based on this same section. And sculptors
and painters down the ages have continued to make use of it.

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The facade of the Parthenon apparently was designed around the proportions of two
large and four small Golden Section, or *5, rectangles, placed above four squares.

This Creek vase, known as a kylix, was designed to be contained within a Golden
Section. The bowl of the vase follows the proportions of four squares placed together
horizontally.

The Golden Mean is a surprising clue to the proportions of the human body. Just look
at the different lengths in your own hand and fingers and forearm, and you can see this
yourself. The length of tile first finger joint is to the length of the next two joints as those
two are to the length of the whole finger! The length of the middle finger is to the length of
the palm as the length of the palm is to the length of the whole hand! The length of the
hand is to the length of the forearm as the length of the forearm is to the whole length
from fingertip to elbow.

Experts have made many more measurements, and have found that this proportion
runs through the whole human skeleton-not exactly, of course, but as a kind of "ideal"
proportion or standard of beauty. That is why the Golden Mean has fascinated some of
the greatest artists through the centuries. Leonardo da Vinci, for instance, called it the
"Divine Section."

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Yet the immediate use of the Golden Mean in geometry, during Plato's time. was even
more remarkable.

The Section was actually the key to the geometric construction of the pentagon and of
the fifth regular solid, the dodecahedron, with its twelve pentagonal faces-not their mere
freehand drawing or building up with tiles as before, but their perfect construction with
string and straightedge. These and other beautiful shapes can be drawn easily if you
just use the Golden Section.

Taking the line AB as radius and using A and C as
centers, draw arcs intersecting at D. Using AC as radius, draw arcs that
cut the long arcs at E and F. Then AC, CF, FD, DE, EA
form a pentagon. A five-pointed star can be formed by drawing AF, EC,
DA, amid DC.

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"But the most important thing about The Section," said Eudoxus to Plato at
dinner, "is the kind of thinking it stimulates. In The Section, the length is irrational,
yet it presents no difficulty because it is handled geometrically. So I have been
working on a new definition of proportion-extending the idea of number to include the
irrationals, and the idea of length so theorems will be correct for all lines...."

Of course, this conversation is imaginary, but Eudoxus of Cnidus actually was the
greatest mathematician of his age, and the Golden Section theorems were his most
striking achievement.

Another great geometer, an Athenian friend of his and Plato's, named Theaetetus,
probably worked on the theorems first, and Plato himself doubtless taught the subject at
the Academy. But it was Eudoxus who finally broke the tyranny of number, with his
magnificent new theory of proportion, so we have made him the hero of our story. He
really must have contemplated the Parthenon in his student days. And we used "poetic
license" to let him demonstrate The Section in the olive grove on the real occasion of his
return visit to Athens. That way, you could see for yourself The Section's brilliant role in
the Golden Age of Greece -and how, at Plato's Academy, Eudoxus and others freed
geometry for the still more brilliant developments to come.